B = (A∩B)⋃(A∩B̅)?

Question

Why is B = (A∩B)⋃(A∩B̅)?
It looks to me like it equals A? This is tripping me up!A∩B is the intersectionA∩B̅ ? Its the intersection of A and everything outside of B, so shouldn't this be A minus its the intersection with B?Then you add them together, it's A? Where am I thinking wrong?

Puzzling · Accepted Answer

You are correct. It should be A, not B.
(A∩B) ⋃ (A∩B̅) = A <--

You can also distribute the intersection to the front and think of this as:
= A ∩ (B⋃B̅)
= A ∩ U <-- the union of B and B̅ is the universal set
= A <-- the intersection of anything with the universal set is just itself.

Here's another analogy. Think of combining all apples that are red with all apples that are not red. The result would be the complete set of apples.

Answer:
Your thinking is correct and the answer of B is a mistake.

B = (A∩B)⋃(A∩B̅)?

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